Mathematics Subject Classification: 26A33, 33E12, 33C20.It has been shown that the fractional integration and differentiation operators transform
such functions with power multipliers into the functions of the same form.
Some of the results given earlier by Kilbas and Saigo follow as special cases

Saigo, Megumi

Saxena, R. K.

Fractional Calculus and Applied Analysis

Bulgarian Digital Mathematics Library at IMI-BAS

CERTAIN PROPERTIES OF FRACTIONAL CALCULUSOPERATORS ASSOCIATED WITHGENERALIZED MITTAG-LEFFLER FUNCTIONR. K. Saxena 1 and Megumi Saigo 2Dedicated to Acad. Bogoljub Stankovic´on the occasion of his 80th birthdayAbstractThis paper deals with the study of an entire function of the formEδβ,γ(z) :=∞∑n=0(δ)nΓ(βn+ γ) n!zn,where β > 0 and γ > 0. For δ = 1, it reduces to Mittag-Leﬄer functionEβ,γ(z). Certain relations that exist between Eδβ,γ(z) and the Riemann-Liouville fractional integrals and derivatives are investigated. It has beenshown that the fractional integration and differentiation operators transformsuch functions with power multipliers into the functions of the same form.Some of the results given earlier by Kilbas and Saigo follow as special cases.Mathematics Subject Classification: 26A33, 33E12, 33C20Key Words and Phrases: Mittag-Leﬄer function, fractional calculus,confluent hypergeometric function, Konhauser polynomials142 R.K. Saxena, M. Saigo1. Introduction and preliminariesThe function defined by the series representationEβ(z) =∞∑n=01Γ(βn+ 1)zn (β > 0, z ∈ C) (1)and its generalizationEβ,γ(z) =∞∑n=01Γ(βn+ γ)zn (β > 0, γ > 0, z ∈ C) (2)were introduced and studied by Mittag-Leﬄer [21, 22], Wiman [25, 26],Agarwal [1], Humbert [10] and Humbert and Agarwal [11], where C is theset of complex numbers. The main properties of these functions are givenin the book by Erde´lyi et al. [4, Section 18.1] and a more extensive anddetailed account on Mittag-Leﬄer functions is presented in Dzherbashyan[2, Chapter 2]. In particular, the functions (1) and (2) are entire functionsof order ρ = 1/β and type σ = 1; see, for example, [2, p.118].The Mittag-Leﬄer function is not given in the tables of Laplace trans-forms, where it naturally occurs in the derivation of the inverse Laplacetransform of the functions of the type pε/(a+ bpβ). This function also oc-curs in the solution of certain boundary value problems involving fractionalintegro-differential equations of Volterra-type [24]. During the various de-velopments of Fractional Calculus in the last three decades this functionhas gained importance on account of its applications in the fields of phys-ical, mathematical and engineering sciences. Hille and Tamarkin [9] havepresented a solution of the Abel-Volterra type equation in terms of Mittag-Leﬄer function. For a detailed account of various properties, generalizationsand applications of this function, the reader may refer to an excellent work ofDzherbashyan [2], Kilbas and Saigo [12, 13, 14, 15], Gorenflo and Mainardi[8], Gorenflo, Luchko and Rogosin [7] and Gorenflo, Kilbas and Rogosin [6].By means of the series representation a generalization of (2) is intro-duced by Prabhakar [23] as:Eδβ,γ(z) =∞∑n=0(δ)nΓ(βn+ γ) n!zn, (3)where β, γ, δ ∈ C (Re(β) > 0). It is an entire function of order [Re(β)]−1[23, p. 7]. It is a special case of Wright’s generalized hypergeometric functionCERTAIN PROPERTIES OF FRACTIONAL CALCULUS . . . 143[27, 28] as well as H-function [5] as shown in (5) and (6), below. For variousproperties of the function defined by (3), see [17].Some important special cases of this function are enumerated below:(i) Eβ(z) = E1β,1(z);(ii) Eβ,γ(z) = E1β,γ(z);(iii) βδEδ+1β,γ (z) = (1 + βδ − γ)Eδβ,γ(z) + Eδβ,γ−1(z);(iv) Φ(γ, δ; z) = Γ(δ)Eγ1,δ(z), where Φ(γ, δ; z) is the Kummer confluenthypergeometric function [3, p. 248, Eq. 1];(v) Zµm(z; k) = Γ(km + µ + 1)E−mk,µ+1(zk), where m, k ∈ R+ = [0,∞) andZµm(·) is a one set of the biorthogonal polynomial pair discussed byKonhauser [19];(vi) If µ ∈ R+, we haveEγm,δ(z) =1Γ(δ) 1Fm(γ;∆(δ;m);zmm), (4)where 1Fm(·) is the generalized hypergeometric function and the sym-bol∆(a;m) represents the sequence of parameters a/m, (a+1)/m, · · · ,(a+m− 1)/m;(vii) Eδβ,γ(z) has the forms:Eδβ,γ(z) =1Γ(δ) 1Ψ1[(δ, 1)(γ, β); z](5)=1Γ(δ)H1,11,2[−z∣∣∣∣ (1− δ, 1)(0, 1), (1− γ, β)](6)=12piωΓ(δ)∫ΩΓ(−s)Γ(δ + s)Γ(γ + sβ)(−z)sds, (7)where 1Ψ1(·) and H1,11,2 (·) are respectively Wright’s generalized hyper-geometric function [27] and H-function [5]. In (7), ω =√−1 and thecontour Ω is a straight line parallel to the imaginary axis separatingthe poles of Γ(−s) at the points s = ν (ν ∈ N0 = {0, 1, 2, · · · }) fromthose of Γ(δ + s) at the points s = −δ − ν (ν ∈ N0). The poles of theintegrand in (7) are assumed to be simple.144 R.K. Saxena, M. SaigoFormula (7) gives the Mellin-Barnes integral representation for the gen-eralized Mittag-Leﬄer function Eδβ,γ(z). A detailed account of H-functionis available from the monographs of Mathai and Saxena [20] and Kilbas andSaigo [16].Another generalization of Wiman function defined by (1) was recentlyintroduced by Kilbas and Saigo [12] in terms of a special entire function ofthe formEα,m,l(z) =∞∑n=0cnzn with cn =n−1∏i=0Γ(α[im+ l] + 1)Γ(α[im+ l + 1] + 1)(n ∈ N0), (8)where an empty product is to be interpreted as unity. Certain propertiesof this function associated with Riemann-Liouville fractional integrals andderivatives are obtained and exact solutions of certain integral equations ofAbel-Volterra type are derived by their applications [12, 14, 15]. The orderand type of the above entire function, defined by (8) alongwith its recur-rence relations, connection with hypergeometric functions and differentialformulas are obtained by Gorenflo, Kilbas and Rogosin [6]. Also, see [6, 8] inthis connection. In a recent paper, Kilbas, Saigo and Saxena [18] obtaineda closed form solution of a fractional generalization of a free electron laserequation of the form:Dατ a(τ) = λ∫ τ0tδa(τ − t)Ebρ,δ+1(iνtρ)dt+ βτσEγρ,σ+1(iντρ) (9)(0 5 τ 5 1)where β, λ ∈ C, ν, b, β ∈ R, α > 0, ρ > 0, σ > −1, δ > −1 and Ebρ,δ+1(z)is the generalized Mittag-Leﬄer function defined by (3). The object of thispaper is to derive the relations that exist between the generalized Mittag-Leﬄer function defined by (3) and the left- and right-sided operators ofRiemann-Liouville fractional calculus [24]. The results derived in this paperare believed to be new.The operators are defined by (see Samko, Kilbas and Marichev [24, Sect.2]) for α > 0: (Iα0+f)(x) =1Γ(α)∫ x0f(t)(x− t)1−α dt; (10)(Iα−f)(x) =1Γ(α)∫ ∞xf(t)(t− x)1−α dt; (11)CERTAIN PROPERTIES OF FRACTIONAL CALCULUS . . . 145(Dα0+f)(x) =(ddx)[α]+1 (I1−{α}0+ f)(x);=1Γ(1− {α})(ddx)[α]+1 ∫ x0f(t)(x− t){α} dt; (12)(Dα−f)(x) =(ddx)[α]+1 (I1−{α}− f)(x);=1Γ(1− {α})(− ddx)[α]+1 ∫ ∞xf(t)(t− x){α} dt, (13)where [α] means the maximal integer not exceeding α and {α} is the frac-tional part of α.2. Properties of generalized Mittag-Leﬄer functionIn this section we derive several interesting properties of the generalizedMittag-Leﬄer function Eδβ,γ(z) defined by (3) associated with Riemann-Liouville fractional integrals and derivatives.Theorem 1. Let α > 0, β > 0, γ > 0 and a ∈ R. Let Iα0+ be the left-sided operator of Riemann-Liouville fractional integral (10). Then thereholds the formula(Iα0+[tγ−1Eδβ,γ(atβ)])(x) = xα+γ−1Eδβ,α+γ(axβ). (14)P r o o f. By virtue of (3) and (10) we haveK ≡(Iα0+[tγ−1Eδβ,γ(atβ)])(x)=1Γ(α)∫ x0(x− t)α−1∞∑n=0(δ)nΓ(βn+ γ) n!antnβ+γ−1dt.Interchanging the order of integration and summation and evaluating theinner integral by beta-function formula, it givesK = xα+γ−1∞∑n=0(δ)nΓ(α+ βn+ γ) n!(axβ)n = xα+γ−1Eδβ,α+γ(axβ).146 R.K. Saxena, M. SaigoThe interchange of the order of integration and summation is permissibleunder the conditions stated with the theorem dues to convergence of theintegrals involved in the process. This completes the proof of Theorem 1.Corollary 1.1. For α > 0, β > 0, γ > 0 and a ∈ R, there holds theformula (Iα0+[tγ−1Eβ,γ(atβ)])(x) = xα+γ−1Eβ,α+γ(axβ). (15)Remark 1. The formula (15) is a known relation [24, Table 9.1, For-mula 23]. If we set β = α in (15), then in view of the relation [4, p. 210,Eq. 23]Eα,γ(x) =1Γ(γ)+ xEα,α+γ(x) (16)it follows that(Iα0+[tγ−1Eα,γ(atα)])(x) =xγ−1a[Eα,γ(axα)− 1Γ(γ)](a 6= 0). (17)We now give a lemma.Lemma 1. For a ∈ R there holds the formulaaxβEδβ,γ(axβ) = Eδβ,γ−β(axβ)− Eδ−1β,γ−β(axβ). (18)P r o o f. The formula (18) is easily verified by virtue of the relation(n+ 1)(δ)n = (δ)n+1 − (δ − 1)n+1.Theorem 1 and Lemma 1 implyTheorem 2. Let α > 0, β > 0, γ > 0, a ∈ R (a 6= 0) and let Iα0+ bethe left-sided operator of Riemann-Liouville fractional integral (10). Thenthere holds the formula(Iα0+[tγ−1Eδβ,γ(atβ)])(x)=1axα+γ−β−1[Eδβ,α+γ−β(axβ)−Eδ−1β,α+γ−β(axβ)]. (19)CERTAIN PROPERTIES OF FRACTIONAL CALCULUS . . . 147Corollary 2.1. For α > 0, β > 0, γ > 0 with α + γ > β and fora ∈ R (a 6= 0), there holds the formula(Iα0+[tγ−1Eβ,γ(atβ)])(x)=1axα+γ−β−1[Eβ,α+γ−β(axβ)− 1Γ(α+ γ − β)]. (20)Remark 2. When β = α in (20), it yields the following result givenby Kilbas and Saigo [14, p. 359, Eq. 20](Iα0+[tγ−1Eα,γ(atα)])(x) =xγ−1a[Eα,γ(axα)− 1Γ(γ)](a 6= 0); (21)(Iα0+[Eα(atα)])(x) =1a[Eα(axα)− 1] (a 6= 0). (22)Theorem 3. Let α > 0, β > 0, γ > 0, a ∈ R and let Iα− be the right-sided operator of Riemann-Liouville fractional integral (11). Then thereholds the formula(Iα−[t−α−γEδβ,γ(at−β)])(x) = x−γEδβ,α+γ(ax−β). (23)P r o o f. By virtue of (3) and (11) we findK ≡(Iα−[t−α−γEδβ,γ(at−β)])(x)=1Γ(α)∫ ∞xt−α−γ(t− x)α−1∞∑n=0(δ)nΓ(βn+ γ) n!ant−βndt.Interchanging the order of integration and summation and evaluating theinner integral, we obtainK =∞∑n=0(δ)nΓ(βn+ α+ γ) n!anx−βn−γ = x−γEδβ,α+γ(ax−β).148 R.K. Saxena, M. SaigoCorollary 3.1. For α > 0, β > 0, γ > 0, a ∈ R there holds theformula (Iα−[t−α−γEβ,γ(at−β)])(x) = x−γEβ,α+γ(ax−β). (24)Theorem 4. Let α > 0, β > 0, γ > 0, a ∈ R (a 6= 0) and let Iα− bethe right-sided operator of Riemann-Liouville fractional integral (11). Thenthere holds the formula(Iα−[t−α−γEδβ,γ(at−β)])(x)=1axβ−γ[Eδβ,α+γ−β(ax−β)− Eδ−1β,α+γ−β(ax−β)]. (25)The proof can be developed on similar lines to that of Theorem 3.Corollary 4.1. For α > 0, β > 0, γ > 0 with α + γ > β and fora ∈ R (a 6= 0) there holds the formula(Iα−[t−α−γEβ,γ(at−β)])(x)=1axβ−γ[Eβ,α+γ−β(ax−β)− 1Γ(α+ γ − β)]. (26)Remark 3. If we set β = α in (26) it reduces to the following resultgiven earlier by Kilbas and Saigo [14, p. 360, Eq. 25], a 6= 0:(Iα−[t−α−γEα,γ(at−α)])(x)=xα−γa[Eα,γ(ax−α)− 1Γ(γ)](a 6= 0);(27)(Iα−[t−α−1Eα(at−α)])(x)=xα−1a[Eα(ax−α)−1](a 6= 0). (28)We now proceed to derive certain other properties of Eδβ,γ(z) associatedwith the fractional derivative operators Dα+0 and Dα− defined by (12) and(13) respectively.CERTAIN PROPERTIES OF FRACTIONAL CALCULUS . . . 149Theorem 5. Let α > 0, β > 0, γ > 0, a ∈ R and let Dα0+ be the left-sided operator of Riemann-Liouville fractional derivative (12). Then thereholds the formula(Dα0+[tγ−1Eδβ,γ(atβ)])(x) = xγ−α−1Eδβ,γ−α(axβ). (29)P r o o f. By virtue of (3) and (12) we haveK ≡(Dα0+[tγ−1Eδβ,γ(atβ)])(x) =(ddx)[α]+1 (I1−{α}0+ [tγ−1Eδβ,γ(atβ)])(x)=∞∑n=0an(δ)nΓ(γ + nβ)Γ(1− {α})n!(ddx)[α]+1 ∫ x0tnβ+γ−1(x− t)−{α}dt=∞∑n=0an(δ)nΓ(γ + nβ + 1− {α})n!(ddx)[α]+1xnβ+γ−{α}=∞∑n=0an(δ)nΓ(nβ + γ − α)n!xγ+nβ−α−1 = xγ−α−1Eδβ,γ−α(axβ),which proves the theorem.Corollary 5.1. For α > 0, β > 0, γ > 0, a ∈ R there holds theformula (Dα0+[tγ−1Eβ,γ(atβ)])(x) = xγ−α−1Eβ,γ−α(axβ). (30)If, however, we set β = α and δ = 1, then (29) also reduces to therelation ([14, p. 362, Eq. 35])(Dα0+[tγ−1Eα,γ(atα)])(x) =xγ−α−1Γ(γ − α) + axγ−1Eα,γ(axα). (31)When γ = 1 in (31) there holds the formula(Dα0+[Eα(atα)])(x) =x−αΓ(1− α) + aEα(axα). (32)Following a similar procedure, we arrive at the following theorem:150 R.K. Saxena, M. SaigoTheorem 6. Let α > 0, γ > β > 0, a ∈ R (a 6= 0) and let Dα0+ be theleft-sided operator of Riemann-Liouville fractional derivative (12). Thenthere holds the formula(Dα0+[tγ−1Eδβ,γ(atβ)])(x)=1axγ−α−β−1[Eδβ,γ−α−β(axβ)− Eδ−1β,γ−α−β(axβ)]. (33)Corollary 6.1. Let α > 0, γ > β > 0, a ∈ R (a 6= 0), then thereholds the formula(Dα0+[tγ−1Eβ,γ(atβ)])(x)=1axγ−α−β−1[Eβ,γ−α−β(axβ)− 1Γ(γ − α− β)]. (34)Theorem 7. Let α > 0, γ > 0 with γ−α+{α} > 1, and a ∈ R, and letDα− be the right-sided operator of Riemann-Liouville fractional derivative(13). Then there holds the formula(Dα−[tα−γEδβ,γ(at−β)])(x) = x−γEδβ,γ−α(ax−β). (35)P r o o f. From (3) and (13) it follows thatK ≡(Dα−[tα−γEδβ,γ(at−β)])(x)=(− ddx)[α]+1 (I1−{α}− [tα−γEδβ,γ(at−β)])(x)=∞∑n=0an(δ)nΓ(nβ + γ)Γ(1− {α})n!(− ddx)[α]+1 ∫ ∞xt−nβ+α−γ(t− x)−{α}dt.If we set t = x/u, then the above expression transforms into the formK =∞∑n=0an(δ)nΓ(nβ + γ)Γ(1− {α})n!×∫ 10unβ−α+γ+{α}−2(1− u)−{α}du(− ddx)[α]+1xα−nβ−γ−{α}+1=∞∑n=0an(δ)nΓ(nβ + γ − α)n! x−nβ−γ = x−γEδβ,γ−α(ax−β).CERTAIN PROPERTIES OF FRACTIONAL CALCULUS . . . 151Corollary 7.1. Let α > 0, γ > β > 0 and a ∈ R, then there holdsthe formula(Dα−[tα−γEβ,γ(at−β)])(x) = x−γEβ,γ−α(ax−β). (36)In a similar manner we can prove the following theorems and corollaries.Theorem 8. Let α > 0, β > 0 with γ − [α] > 1, a ∈ R (a 6= 0) and letDα− be the right-sided operator of Riemann-Liouville fractional derivative(13). Then there holds the formula(Dα−[tα−γEδβ,γ(at−β)])(x)=1axβ−γ[Eδβ,γ−α−β(ax−β)−Eδ−1β,γ−α−β(ax−β)]. (37)Corollary 8.1. For α > 0, β > 0 with γ − [α] > 1, a ∈ R (a 6= 0) letα+ γ > β, there holds the formula(Dα−[tα−γEβ,γ(at−β)])(x)=1axβ−γ[Eβ,γ−α−β(ax−β)− 1Γ(γ − α− β)]. (38)Theorem 9. Let α > 0, β > 0 with γ − [α] > 0, a ∈ R (a 6= 0) and letDα− be the right-sided operator of Riemann-Liouville fractional derivative(13). Then there holds the formula(Dα−[tα−γ−1Eδβ,γ(at−β)])(x)=1ax−γ−1[(γ − βδ)Eδβ,γ−α+1(ax−β) + βδEδ+1β,γ−α−1(ax−β)]. (39)152 R.K. Saxena, M. SaigoCorollary 9.1. Let α > 0, β > 0 with γ− [α] > 0 and a ∈ R (a 6= 0),then there holds the formula(Dα−[tα−γ−1Eβ,γ(at−β)])(x)=1ax−γ−1[(γ − β)Eβ,γ−α+1(ax−β) + βE2β,γ−α+1(ax−β)]. (40)References[1] R.P. Agarwal, A propos d’une Note M. Pierre Humbert. C. R. Acad.Sci. Paris 236(1953), 2031-2032.[2] M. M. Dzherbashyan, Integral Transforms and Representations of Func-tions in Complex Domain (Russian). Nauka, Moscow (1966).[3] A. Erde´lyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, HigherTranscendental Functions, Vol. I. McGraw-Hill, New York-Toronto-London (1953).[4] A. Erde´lyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, HigherTranscendental Functions, Vol. III. McGraw-Hill, New York (1955).[5] C. Fox, The G andH-functions as symmetrical Fourier kernels. Trans.Amer. Math. Soc. 98 (1961), 395-429.[6] R. Gorenflo, A.A. Kilbas and S.V. Rogosin, On the generalized Mittag-Leﬄer type function. Integral Transforms Spec. Funct. 7 (1998),215-224.[7] R. Gorenflo, Yu. Luchko and S.V. Rogosin, Mittag-Leﬄer type func-tions, notes on growth properties and distribution of zeros. PreprintNo. A04-97, Freie Universita¨t Berlin, Serie A Mathematik, Berlin(1997).[8] R. Gorenflo and F. Mainardi, The Mittag-Leﬄer type function in theRiemann-Liouville fractional calculus. In: Boundary Value Problems,Special Functions and Fractional Calculus (Proc. Int. Conf. Minsk1996) Belarusian State Univ., Minsk (1996), 215-225.[9] E. Hille and J.D. Tamarkin, On the theory of linear integral equations.Ann. Math. 31 (1930), 479-528.CERTAIN PROPERTIES OF FRACTIONAL CALCULUS . . . 153[10] P. Humbert, Quelques resultats d’le fonction de Mittag-Leﬄer. C. R.Acad. Sci. Paris 236 (1953), 1467-1468.[11] P. Humbert and R.P. Agarwal, Sur la fonction de Mittag-Leﬄer etquelques unes de ses generalizations. Bull. Sci. Math. (2) 77 (1953),180-185.[12] A.A. Kilbas and M. Saigo, On solution of integral equations of Abel-Volterra type. Differential and Integral Equations 8 (1995), 993-1011.[13] A.A. Kilbas and M. Saigo, Fractional integrals and derivatives ofMittag-Leﬄer type function (Russian). Dokl. Akad. Nauk Belarusi39, No 4(1995), 22-26.[14] A.A. Kilbas and M. Saigo, On Mittag-Leﬄer type function, fractionalcalculus operators and solutions of integral equations. Integral Trans-form. Spec. Funct. 4 (1996), 355-370.[15] A.A. Kilbas and M. Saigo, Solution in closed form of a class of lineardifferential equations of fractional order (Russian). DifferentsialnyeUravnenija 33 (1997), 195-204; Translation in: Differential Equations33 (1997), 194-204.[16] A.A. Kilbas and M. Saigo, H-Transforms. Theory and Applications.Ser. ’Analytic Methods and Special Functions’, Vol. 9, CRC Press,London and New York (2004).[17] A.A. Kilbas, M. Saigo and R.K. Saxena, Generalized Mittag-Leﬄerfunction and generalized fractional calculus operators. Integral Trans-form. Spec. Funct. 15 (2004), 31-49.[18] A.A. Kilbas, M. Saigo and R.K. Saxena, Solution of Volterra integro-differential equations with generalized Mittag-Leﬄer function in thekernels. J. Integral Eq. Appl. 14 (2002), 377-396.[19] J.D.E. Konhauser, Biorthogonal polynomials suggested by the La-guerre polynomials. Pacific J. Math. 21 (1967), 303-314.[20] A.M. Mathai and R.K. Saxena, The H-Function with Applications inStatistics and Other Disciplines. John Wiley and Sons, New York-London-Sydney (1978).[21] G.M. Mittag-Leﬄer, Sur la nouvelle fonction Eα(x),˙C. R. Acad. Sci.Paris 137 (1903), 554-558.154 R.K. Saxena, M. Saigo[22] G.M. Mittag-Leﬄer, Sur la representation analytiqie d’une fonctionmonogene (cinquieme note). Acta Math. 29(1905), 101-181.[23] T.R. Prabhakar, A singular integral equation with a generalized Mittag-Leﬄer function in the kernel. Yokohama Math. J. 19(1971), 7-15.[24] S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals andDerivatives. Theory and Applications. Gordon and Breach, Yverdonet al. (1993).[25] A. Wiman, U¨ber den Fundamental satz in der Theorie der FunctionenEα(x). Acta Math. 29 (1905), 191-201.[26] A. Wiman, U¨ber die Nullstellun der Funktionen Eα(x). Acta Math.29(1905), 217-234.[27] E.M. Wright, The asymptotic expansion of the generalized hypergeo-metric functions. J. London Math. Soc. 10(1935), 286-293.[28] E.M. Wright, The asymptotic expansion of the generalized hyperge-ometric functions. Proc. London Math. Soc. (2) 46(1940), 389-408.1 Department of Mathematics and StatisticsJai Narain Vyas University Received: June 5, 2003Jodhpur - 342001, INDIAe-mail: ravisax@sancharnet.in(Postal address: 34 Panch Batti ChaurahaRatanada, Jodhpur - 342011, INDIA)2 Department of Applied MathematicsFukuoka UniversityFukuoka 814-0180, JAPANe-mail: msaigo@fukuoka-u.ac.jp

Certain Properties of Fractional Calculus Operators Associated with Generalized Mittag-Leffler Function

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Certain Properties of Fractional Calculus Operators Associated with Generalized Mittag-Leffler Function

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